Teak Chairs

I chair some old teak interior that are black in some parts of the use and just need a good cleaning. What is the best way to do it?

Try a marine supply store for teak cleaners and finishes used on boats ..
One consideration is that the preparations can be walked on Navy w / bare feet. (And others)
Some cleaners may need to be neutralized, so read the box.
Teak is difficult and will withstand a light sanding if absolutely needed.
Once cleaned and dry you can apply a finish to the marine oil or marine varnish if you want a bright shine.
Either can be rubbing in rags w / or bare hands.
Guessing about $ 25 for small amounts (even good things) you'll need for a couple of chairs.
Cordially

Try Murphy Oil Soap (found in local grocery stores or mass merchants) and a pot scrubber anti-scratch (it's like steel wool, plastic). This should take the dirt, dust and grime without harming the natural finish.

oil soap use Murfy